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AP Physics B1
Chapter 8- Rotational Motion
Questions for Discussion
Angular Displacement, Velocity, Acceleration, Angular Kinematics, Roll Without Slip (Sec 1-16)
1. A wheel accelerates at 1.2 rad /s2 for 24 s. The angular displacement of the wheel is 640 rad
during the 24 s. The radius of the wheel is .90 m.
(a) What was the initial angular velocity of the wheel in rad/s?
(b) What was the final angular velocity of the wheel in rad/s?
(c) What was the
€ final linear speed of a stone caught in the tread of the wheel?
(d) What was the linear (tangential) acceleration of a stone caught in the tread of the wheel?
(e) What is the final centripetal acceleration of a stone caught in the tread of the wheel?
2. A merry-go-round rotates at the rate of .30 rad/s. The merry-go-round is slowed down by the
force of friction and accelerates at the rate of −.012 rad /s2 .
(a) How long will it take the merry-go-round to come to rest?
(b) How many revolutions will the merry-go-round make while it is coming to rest?
3. A flywheel of radius .30 m starts from
€ rest and accelerates with constant angular acceleration
2
of .50 rad /s . Compute the tangential acceleration, the radial acceleration and the resultant
acceleration (magnitude and angle the resultant acceleration makes with the radial direction) of
a point on its rim at the following times.
(a) at the start of the motion
€ (b) just after the flywheel has turned through 120°
4.
Q
P
O
The disk rotates about stationary axis O. Point Q is twice as far from the axis as point P. The
angular acceleration of the disk is constant and its angular speed at the instant shown is ω .
(a) The angular speed of point Q is (less than, equal to, greater than) the angular speed of
point P.
(b) The angular acceleration of point Q is (less than, equal to, greater than) the angular speed
of point P.
€
(c) The linear speed of point Q is ___ times the linear speed of point P.
(d) The magnitude of the tangential acceleration of point Q is ___ times that of point P.
(e) The magnitude of the radial acceleration of point Q is ___ times that of point P.
(f) The magnitude of the linear acceleration of point Q is ___ times that of point P.
(g) The angle that the linear acceleration of point Q makes with the radial direction is ___ times
the angle that the linear acceleration of point P makes with the radial direction.
(h) The angular displacement of point Q in a given time ∆ t is (less than, equal to, greater than)
the angular displacement of point P in the same time.
(i) The distance point Q travels in a given time ∆ t is (less than, equal to, greater than)
the distance point P travels in the same time.
5. A disk is rotating about a stationary axis. Consider point P on the rim of the disk.
(a) Under what conditions will the linear velocity of point P be parallel to its tangential
acceleration?
(b) Under what conditions will the linear velocity of point P be perpendicular to its
(total) linear acceleration?
(c) Under what conditions will the linear velocity of point P be parallel to its (total) linear
acceleration?
6. A disk is rotating about a stationary axis. Point P is on the rim of the disk. The angular
velocity of the disk is positive (counterclockwise rotation) but the angular acceleration is
constant and negative so that the rotation of the disk is slowing. Consider the time the rotation
of the disk is remains counterclockwise.
(a) The magnitude of the tangential acceleration of point P (decreases, remains the same,
increases) with time.
(b) The magnitude of the radial acceleration of point P (decreases, remains the same, increases)
with time.
7. A disk rotates from rest and angular position θ = 0. The angular acceleration of the disk is
constant. Point P is a point on the rim of the disk. The magnitude of the tangential
acceleration of point P is 3.0 m /s2 and the magnitude of the radial acceleration of point P
is 7.0 m /s2 at the instant the angular position of the disk is θ 0 .
€
(a) What is the magnitude of the tangential
acceleration of point P at the instant the angular
position of the disk is 2 θ 0 ?
€
(b) What is the magnitude
of the radial acceleration of point P at the instant the angular
€
position of the disk is 2 θ 0 ?
€
8. A driver floors the€accelerator of his car when starting from rest at a traffic light. The tires
screech as they spin but the car barely moves forward. Let Δθ be the angle that the tires turn
while they screech€and s be the linear distance that the car travels. The radius of the tires is r.
The distance s is (less than, equal to, greater than) r Δθ .
€
9. A driver hits the brakes while traveling at high speed.
The car skids. Let Δθ be the angle that
the tires turn while the car skids and s be the linear distance that the car travels. The radius of
€
the tires is r. The distance s is (less than, equal
to, greater than) r Δθ .
10. A car travels at constant speed. Let Δθ be the angle that the tires€turn while the car moves
and s be the linear distance that the car travels. The radius of the tires is r. The distance s is
€
(less than, equal to, greater than) r Δθ .
11. A bicycle wheel has radius€.35 m. The wheel starts from rest at time t= 0 and accelerates
uniformly to a speed of 9.0 rad/s at time t= T. The bicycle rolls without slip.
(a) What is the speed of€the bicycle at time T?
(b) How far does the bicycle travel from t= 0 to time T? Answer in terms of T.
P
12.
v
Q
O
A bicycle tire rolls without slip. The speed of the center O of the tire is v. Point Q is directly
to the left of point O and point P is directly above point O at the instant shown.
(a) What is the speed of point Q with respect to the ground at the instant shown?
(b) What is the speed of point P with respect to the ground at the instant shown?
13. The sensor in a speedometer measures the angular speed with which a tire rotates. The
assumed radius of the tire is used to convert that angular speed into a linear speed that the
speedometer displays. If a car’s tires are very worn then the speedometer reading will
be (less than, equal to, greater than) the car’s actual speed.
14. A grinding wheel starts from rest and accelerates about a fixed axis at constant angular
acceleration. The displacement of the wheel from t= 0 to t= T is Δθ and the angular speed of
the wheel at t= T is ω .
(a) What is the angular speed of the wheel at t= 2 T? Answer in terms of ω .
(b) What is the angular displacement of the wheel from t= 0 to t= 2 T? Answer in terms
of Δθ .
€
€
Torque and Rotational Inertia (Sections 17 to 21)
€
15.
€ A motor’s axle passes through the center O of a disk. Two masses are glued to the disk.
Case 1
Case 2
O
O
(a) The rotational inertia of the system in Case 2 is (smaller than, the same as, larger than) the
rotational inertia in Case 1.
(b) Assuming the motor applies the same torque in both cases, the angular acceleration of the
system in Case 2 is (smaller than, the same as, larger than) the angular acceleration
in Case 1.
16. A piece of clay has mass M. The clay is molded into a square of uniform thickness. The
moment of inertia of the square about an axis perpendicular to the plane of the square and
passing through its center is I.
(a) Suppose the clay is flattened so that a larger square of uniform thickness is formed. The
sides of the new square are three times as long as the sides of the original square. What is
the rotational inertia of the new square about the same axis as before?
(b) Suppose a new square is formed by adding clay to the old square. The new square has the
same thickness as the old one. The sides of the new square are three times as long as the
sides of the original square. What is the moment of inertia of the new square about the
same axis as before?
Note:
A square plate has sides of length a and mass m. The axis of rotation of the square plate is
perpendicular to the plane of the square and passes through its center. The moment of inertia
1
of the square plate is given by I =
ma2 .
6
17. A clay cylinder has radius R, height h and mass M. The moment of inertia about an axis
passing through the centers of the ends of the cylinder is I.
(a) A new cylinder with
€ twice the mass but the same radius is formed by adding clay to the
old cylinder. What is the moment of inertia of the new cylinder? The axis of rotation is as
before.
(b) A new cylinder with twice the mass but the same height h is formed by adding clay to the
original cylinder (not the cylinder of part (a)). What is the moment of inertia of the new
cylinder about the same axis as before?
Note:
The rotational inertia I of a cylinder of radius R and mass M is the same as that of a disk of
radius R and mass M. The intersection of a plane that is perpendicular to the axis of a cylinder
or a disk is a circle. Both shapes thus have mass distributed the same way about the axis.
1
For both shapes I= MR2 .
2
18.
8.0 N
12 N
€
O
30 °
2.0 m
3.0 m
Calculate €
the resultant torque about point O for the two forces applied to the rod shown above.
Take counterclockwise as the positive direction.
19.
X
3R
2R
F
F
2F
The object shown above consists of two disks that are glued together so that they rotate freely
about an axis that passes through their centers and is perpendicular to the page. If the net
torque on the object is zero then it remains at rest. What is the magnitude of the force,
labeled X, that is required for equilibrium? Express your answer in terms of F and R.
20. If the net force on an object is zero then must the net torque on the object also be zero? If not
then give a counterexample.
21. If the net torque on an object is zero then must the net force on the object also be zero? If not
then give a counterexample.
22. The axis of a disk is perpendicular to its plane. The axis passes through a point that is part of
the disk. Where should the axis be located so that the rotational inertia of the disk about the
axis is as large as possible?
23.
F
F
Rope
A man is trying to remove a bolt using a wrench. He is unable to turn the wrench. He then ties
a rope to the wrench as shown above and pulls on the rope with the same force as before. Will
this strategy help the man remove the bolt? Explain.
24. Why would a man attempting to walk across the Grand Canyon on a rope stretched across it
carry a long pole?
(a) increase the torque that acts on him
(b) increase his potential energy
(c) increase his rotational inertia
(d) increase his weight
(e) increase the height of his center of mass
Explain your reasoning.
25. The net torque on a rotating wheel is τ and the wheel turns at angular speed ω . What power is
developed by the torque?
Hint:
Start with a power formula from linear motion and substitute the analogous rotational
quantities.
€
€
26.
F
Pivot
A
d
Pivot
B
d
2F
30°
d
d
2F
Pivot
60°
C
d
d
2F
Pivot
30°
D
d
d
F
Pivot
30°
E
d
d
Pivot
3F
F
d
d
A force acts on a bar that is free to pivot about its left end in each case. Rank the magnitudes
of the torque that the forces produce by assigning a number to each case. Use 1 for the
smallest magnitude and larger numbers for larger magnitudes. If two or more cases have a
torque with the same magnitude then assign the same number to such cases.
A ____
B ____
C ____
D ____
E ____
F____
27.
Cylinder B
ρ
Density
2
Cylinder A
Density ρ
Rotational Inertia I
Axis
Axis
€
€
R
R
M
M
Solid cylinders A and B have the same mass and the same radius. Cylinder B is made of a less
dense material than cylinder A. Express the rotational inertia of cylinder B in terms of I, the
rotational inertia of cylinder A.
28. A solid sphere has mass M and radius R. A hollow shell has mass M and radius R. Which
object has the greater rotational inertia about an axis passing through its center? Explain.
29.
Axis
M
L
M
A system consists of two very small objects of mass M connected by a rigid rod of length L
and negligible mass. John wishes to calculate the rotational inertia of the system about an axis
going through the object on the left. John applies the formula I= mR 2 to the center of mass of
⎛ L ⎞ 2 1
the object and obtains the result I = (2M)⎜ ⎟ = ML 2 .
⎝ 2 ⎠
2
(a) Is John correct? If not, why?
€ rotational inertia I for the system
(b) If John is not correct then what is the correct value of
about the given axis?
€
Rotational Dynamics Problems (Sections 22 to 25)
€
30. The rotational inertia of a disk about an axis through its center and perpendicular to its plane is
1
I=
mr 2 . A grinding disk has radius .20 m and mass 1.6 kg. The disk is spinning
2
at 15 rev/s. A tangential frictional force of 8.0 N acts due to an object pressed against the rim
of the disk.
(a) What is the angular acceleration of the disk?
(b) How long does it take the disk to come to rest?
(c) How many revolutions does the disk make while it is slowing down?
31. A potter’s wheel has mass 80 kg and radius 2.0 m.
1
The rotational inertia I of a disk is given by I =
mr 2 .
2
What torque is required
to
make
the
wheel
spin
at
40 rev/min if the wheel starts from rest
€
and reaches 40 rev/min after making 20 rev?
32. A uniform rod 2.0 m long and of €
mass 3.6 kg is hinged on its left side. The rotational inertia
€ 1
of€
a rod hinged on one end €
is given by I =
ML 2 , where M is the mass of the rod and L is
3
its length. What is the angular acceleration of the rod at the instant it is released from a
horizontal position?
33. A bucket of water of mass 20€kg is suspended by a rope wrapped around a windlass in the
form of a solid cylinder .20 m in diameter, also of mass 20 kg. The cylinder is pivoted on a
frictionless horizontal axle that passes through the centers of its circular ends. The bucket is
released from rest at the top of the well and falls 20 m to the water. Neglect the weight of the
€
rope.
€
(a) What is the tension in the rope while the bucket
is falling?
(b) How long did it take the bucket to fall to the water?
€
34.
L
3M
P
2M
A light rod has two masses attached to its ends, as shown. If the rod is free to pivot about an
axis perpendicular to P then what is the initial angular acceleration of the rod? The distance
from the left end of the rod to P is L/4. The rod is horizontal when it is released from rest.
Answer in terms of L.
35.
O
5.0 kg
37°
A block of mass 5.0 kg slides down a surface inclined 37° to the horizontal, as shown above.
The coefficient of sliding friction is .25 . A string attached to the block is wrapped around a
flywheel on a fixed axis at O. The flywheel has a mass M= 20 kg, an outer radius of .20 m
and a moment of inertia with respect to the axis of .20 kg•m 2 .
(a) What is the acceleration of the block down the plane?
(b) What is the tension in the string?
€
36. As the rod of problem 32 swings toward the €
vertical, the magnitude of its angular acceleration
(increases, decreases, remains the same). Explain.
Rotational Kinetic Energy, Conservation of Energy and Rolling Without Slip (Sections 26-29)
37. Four very small spheres, each of mass 3.0 kg are arranged in a square .50 m on a side and are
connected by light rods of negligible mass. Find the rotational inertia of the system about an
axis through the center of the square, perpendicular to its plane.
38. A uniform meter stick of mass .40 kg is pivoted about one end so it can rotate without friction
about a horizontal axis. The meter stick is held in a horizontal position and released. Calculate
each of the following at the instant the meter stick swings through the vertical.
(a) the angular velocity of the meter stick
(b) the linear velocity of the end of the meter stick opposite the axis
Hint:
You can calculate gravitational potential energy by treating the mass of an object as if it were
concentrated at the center of mass.
1
The rotational inertia of a uniform rod that is pivoted about one end is I =
ML 2 .
3
€
39.
4.0 kg
5.0 m
2.0 kg
Consider the system shown above. The pulley has radius .20 m and rotational
inertia .32 kg•m 2 . The rope does not slip on the pulley rim. The system is released from rest.
Use energy methods to calculate the velocity of the 4.0 kg block just before it strikes the floor.
40.
Block A
€
mA
I
€
mB
Block B
Consider the system shown above. The pulley€has radius R and rotational inertia I. The rope
does not slip over the pulley. The coefficient of friction between block A and the table top
is µK . The system is released from rest and block B descends. Use energy methods to write
an equation that relates m A , m B, I, R, µK , the linear speed v of the blocks and the distance d
that block B has fallen.
€
€
€
€
41. A hoop rolls on a horizontal surface without slipping. What fraction of the total kinetic energy
of the hoop is rotational kinetic energy? The rotational inertia of a hoop about its center is
given by I = MR2 .
42.
€
h
θ
A solid sphere of mass M and radius R rolls down an inclined plane without slipping. Answer
each of the following in terms of M, R, g,€θ and h (h is the distance from the sphere’s lowest
point to the base of the plane). The sphere is released from rest at the position shown and rolls
2
without slip. The rotational inertia of a sphere about its center is given by I =
MR2 .
5
€
(a) What is the sphere’s linear speed
when it reaches the bottom of the incline?
(b) What is the sphere’s total kinetic energy when it reaches the bottom of the incline?
(c) What is the sphere’s linear (translational) kinetic energy when it reaches the bottom of
the plane?
€
(d) What is the sphere’s rotational kinetic energy when it reaches the bottom of the plane?
In parts (e) and (f), the solid sphere is replaced by a hollow sphere of the same mass M and
radius R. The hollow sphere is released from rest at the original location of the solid sphere.
The hollow sphere rolls without slipping. Justify your answer to each question.
(e) The total kinetic energy of the hollow sphere is (less than, equal to, greater than) the total
kinetic energy of the solid sphere at the bottom of the plane.
(f) The rotational kinetic energy of the hollow sphere is (less than, equal to, greater than) the
rotational kinetic energy of the solid sphere at the bottom of the plane.
43. Find the magnitude of the linear acceleration of the sphere in problem 42.
Hints:
Draw a free body diagram of the sphere.
Use the diagram to write an equation using ΣF = Ma in the direction parallel to the incline.
Also write an equation using Στ = Iα . Use the center of the sphere as the pivot point.
Do not replace F FRIC with µF N . Instead, substitute for F FRIC in the first equation using an
expression from the second equation.
€
€
€
€
€
44.
F
A lawn roller in the form of a hollow cylinder of mass M is pulled horizontally with constant
force F applied by a handle attached to the axle. If it rolls without slipping then find the
acceleration of the roller and the friction force that act on it. Answer in terms of F and M.
The rotational inertia of the hollow cylinder (the circular ends of the cylinder have negligible
mass) is given by I = MR2 . Use the technique outlined in the hints for problem 43.
45. Suppose the solid sphere of problem 42 rolls with slip as it moves down the incline because
there is not enough static friction between the sphere and the incline.
(a) When
€ the sphere reaches the bottom of the incline, its rotational kinetic energy will be
(less than, equal to, greater that) the rotational kinetic energy of the sphere in the no slip
case. Explain.
Hint:
Consider the extreme case in which there is no friction at all.
(b) When the sphere reaches the bottom of the incline, its total mechanical energy will be
(less than, equal to, greater than) the total mechanical of the sphere in the no slip case.
Explain.
Hint:
Consider the argument of section 27 in the notes. Which will be greater- the amount by
which sliding friction decreases the translational kinetic energy of the sphere or the amount
by which sliding friction increases the rotational kinetic energy of the sphere?
(c) Kinetic friction does (negative, zero, positive) work in the rolling with slip case.
46. Suppose there were no friction at all between the solid sphere of problem 42 and the incline.
What would be the linear speed of the sphere at the bottom of the incline in this case?
47. An object rolls without slip from rest down an incline. The speed of the object at the bottom of
the incline depends on which of the following factors?
(a) the mass of the object
(b) the radius of the object
(c) the rotational inertia of the object
(d) the angle that the incline makes with the horizontal
48. The rotational inertia of a sphere about its center is given by I =
2
MR2 .
5
1
MR2 .
2
The rotational inertia of a hoop about its center is given by I = MR2 .
€ radius. Suppose each object has linear
A sphere, disk and hoop all have the same mass and
speed v at the bottom of a long incline and that each object rolls up the incline without slip.
List the objects in order of how far they roll up the
€ incline before beginning to roll back down.
List the object that rolls the smallest distance first.€
The rotational inertia of a disk about its center is given by I =
49. Does the force of friction in problem 48 act parallel to the incline and toward the top of the
incline or parallel to the incline and toward the bottom as each object rolls toward the top?
Hint:
What happens to each object’s angular speed as it rolls toward the top of the incline? Does the
torque due to friction act in the same sense as the rotation of the object or does the torque
oppose the rotation of the object?
50. Two soup cans roll down an incline from rest without slip. Both cans have the same radius,
same mass and start at the same height. Can A is filled with a watery soup. There is very little
friction between the walls of the can and the soup. Can B is filled with a thick soup. There is
much friction between the walls of the can and the soup. Which can has the greater linear
speed when it reaches the bottom of the incline? Why?
51. Ball A has mass M and radius R. It rolls without slip down an incline from vertical height H.
Ball B has mass 2M and radius R. It rolls without slip down an identical incline from vertical
height H. Both balls start from rest. The rotational inertia of a sphere about its center is given
2
by I =
MR2 .
5
(a) The rotational inertia of ball B is ___ times the rotational inertia of ball A.
(b) The speed of ball B at the bottom of its incline is ___ times the speed of ball A at the
bottom of its incline.
(c)
The translational kinetic energy of ball B at the bottom of its incline is ___ times the
€
translational kinetic energy of ball A at the bottom of its incline.
(d) The total kinetic energy of ball B at the bottom of its incline is ___ times the
total kinetic energy of ball A at the bottom of its incline.
(e) The angular speed of ball B at the bottom of its incline is ___ times the angular speed of
ball A at the bottom of its incline.
(f) The rotational kinetic energy of ball B at the bottom of its incline is ___ times the
rotational kinetic energy of ball A at the bottom of its incline.
(g) The time it takes ball B to travel down its incline is ___ times the time it takes ball A to
travel down its incline.
52. Ball A has mass M and radius R. It rolls without slip down an incline from vertical height H.
Ball B has radius 2R and is made of the same material as ball A. It rolls without slip down an
identical incline from vertical height H. Both balls start from rest. The rotational inertia of a
2
sphere about its center is given by I =
MR2 .
5
(a) The rotational inertia of ball B is ___ times the rotational inertia of ball A.
(b) The speed of ball B at the bottom of its incline is ___ times the speed of ball A at the
bottom of its incline.
(c) The translational kinetic
€ energy of ball B at the bottom of its incline is ___ times the
translational kinetic energy of ball A at the bottom of its incline.
(d) The total kinetic energy of ball B at the bottom of its incline is ___ times the
total kinetic energy of ball A at the bottom of its incline.
(e) The angular speed of ball B at the bottom of its incline is ___ times the angular speed of
ball A at the bottom of its incline.
(f) The rotational kinetic energy of ball B at the bottom of its incline is ___ times the
rotational kinetic energy of ball A at the bottom of its incline.
(g) The time it takes ball B to travel down its incline is ___ times the time it takes ball A to
travel down its incline.
Hint:
4
The mass of ball B is proportional to its volume. The volume of a sphere is
π R 3 . Thus, the
3
mass of ball B is proportional to the cube of its radius.
53. A hoop has mass M and radius R. A disk has mass M and radius R. Both objects roll from
rest down identical inclines without slip. The rotational inertia of a€disk about its center is
1
given by I =
MR2 . The rotational inertia of a hoop about its center is given by I = MR2 .
2
(a) The total kinetic energy of the disk at the bottom of its incline is ___ times the total
kinetic energy of the hoop at the bottom of its incline.
(b) The speed of the disk at the bottom of its incline is ___ times the speed
€ of the hoop at
the
bottom
of
its
incline.
€
(c) The angular speed of the disk at the bottom of its incline is ___ times the angular speed
of the hoop at the bottom of its incline.
(d) The translational kinetic energy of the disk at the bottom of its incline is ___ times the
translational kinetic energy of the hoop at the bottom of its incline.
(e) The rotational inertia of the disk is ___ times the rotational inertia of the hoop.
(f) The rotational kinetic energy of the disk at the bottom of its incline is ___ times the
rotational kinetic energy of the hoop at the bottom of its incline.
1
Hint: Combine the results of parts (c) and (e) with the formula K ROT =
I ω2.
2
€
54.
R
m
d
You have designed an experiment to measure the rotational inertia of a disk. The disk is used
as a pulley in the arrangement shown above. String is wrapped around the disk several times
and a mass m is tied to the other end of the string. The mass is released from rest and the
distance d that the mass falls is measured as a function of the time t that elapses after the mass
is released.
time t (s)
distance d (m)
.20
.063
.40
.25
.60
.54
.80
.92
1.0
1.45
Vertical
Horizontal
Use the data you have collected to find the magnitude a of the acceleration of the mass m.
Do so by making a graph of the data that gives a straight line whose slope equals the
magnitude of the acceleration.
(a) Complete the last two columns of the table. In the top row write an expression that
indicates how a column of data will be transformed so that when the transformed data are
plotted a straight line whose slope is a will result. For example, if you need to multiply
each distance by five your expression would be 5 d. Fill in the rest of the rows of the
table with the numerical values of your transformed data.
(b) Plot your data on the graph.
€
Include an appropriate scale for each axis.
Label each axis, including units.
Draw a best fit line through the plotted points.
(c) What is your experimental value for the magnitude a of the acceleration of mass m?
⎞
2 ⎛ g
(d) Show that the rotational inertia I of the pulley can be written I = mR ⎜ – 1⎟ .
⎝ a
⎠
Include free body diagrams of the mass and the pulley in your work.
(e) Use the values m= 1.5 kg, R= .40 m, g= 9.80 m /s2 and your experimental value of a to
find an experimental value of the rotational inertia €
I of the disk.
(f) The accepted value of the rotational inertia of the disk is .60 kg • m2 . Calculate the percent
error for your experimental value of€rotational inertia.
(g) State a likely reason why your experimental value of rotational inertia is too small.
€
Angular Momentum and Its Conservation (Sections 30 to 32)
55.
(0, b)
O
(a, 0)
(c, d)
m
A particle travels along a line with x intercept a and y intercept b. The mass of the particle
is m, its speed is v and its coordinates are (c, d). What is the magnitude of the particle’s
angular momentum about origin O?
Hint:
Angular momentum is constant. The altitude to the hypotenuse of a right triangle times
hypotenuse equals the product of the legs.
56. A solid wood door 1.0 m wide and 2.0 m high is hinged along one side and has a total mass
of 50 kg. Initially open and at rest, the door is struck at its center with a hammer. During the
blow an average force of 2000 N acts for .010 s. Find the angular velocity of the just door
after impact. The rotational inertia of a bar pivoted about one of its ends is given by
1
I=
ML 2 . The intersection of a plane perpendicular to the hinged side of the door and the
€
3
€ in each cross section of a door is distributed in the same way as the mass
door is a rod. Mass
is distributed in a rod. So rotational inertia of a door is calculated using the formula for the
rotational inertia of a rod pivoted about one of its ends.
€
57.
A puck on a frictionless air hockey table has a mass of .050 kg and is attached to a cord
passing through a hole in the table surface, as shown above. The puck is originally revolving at
a distance of .20 m from the hole, with an angular velocity of 3.0 rad/s. The cord is then pulled
from below, shortening the radius of the circle in which the puck revolves to .10 m. The puck
may be considered a point mass.
(a) What is the new angular velocity?
(b) Find the change in kinetic energy of the puck.
58.
TOP
VIEW
O
w
m
v
A bullet has mass m. The bullet strikes a door whose mass is M at its unhinged edge. The
initial velocity v of the bullet is perpendicular to the door. The bullet becomes embedded in the
door. The height of the door is h and its width is w . Ignore friction in the hinges.
1
Use I= Mw 2 to find the rotational inertia of the door.
3
(a) What is the angular velocity of the door just after the impact of the bullet? Answer in terms
of M, m, w and v.
(b) Do you think linear momentum was conserved in this collision? Explain.
€ (c) Do you think mechanical energy was conserved in this collision? Explain.
59.
Before
Collision
After
Collision
v
v
v
v
Each system interacting in the diagram consists of two very small objects, each of mass m. The
objects are joined by a rigid rod of negligible mass. The systems move on a horizontal
frictionless surface. The motion before and after a collision of two of these systems is shown.
The collision illustrated would violate which of these laws?
(a) Conservation of Linear Momentum
(b) Conservation of Energy
(c) Conservation of Angular Momentum
60. Sphere A and sphere B each rotate about a stationary axis passing through its center. The
magnitude of each sphere’s angular momentum is the same. Sphere B has twice as much
rotational kinetic energy as sphere A. The rotational inertia of sphere B is ___ times the
rotational inertia of sphere A.
61. No external forces act on an isolated system. Which of the following must remain constant?
(a) velocity of the center of mass of the system
(b) translational kinetic energy of the system
(c) total kinetic energy of the system
(d) magnitude of the linear momentum of the system
(e) magnitude of the angular momentum of the system
62. Give an example of an object whose angular momentum is conserved although a net force acts
on the object.
63. A car has rear wheel drive. The center of mass of the car rises so that the front end tips upward
when the car accelerates from rest at a traffic light.
(a) What force produces the torque that tends to turn the body of the car about the rear axle?
(b) Would the front end of the car still tip upward if the car had front wheel drive?
(c) Would the front end of the car still tip upward if the car had four wheel drive?
(d) Which way would the front end of the car tip if the car suddenly stopped while moving
forward?
(e) Which way would the front end of the car tip if the car suddenly stopped while moving
backward?
(f) Suppose the car were “floating” in the cargo bay of a space station. Would the body of
the car turn if the accelerator were pushed so that the wheels started to spin?
64.
v2
Sun
v1
r1
r2
€
€
€ is in an elliptical
orbit about the Sun. The comet’s distance from the Sun is a
€A comet
maximum in position 2 and a minimum in position 1. The speed of the comet is v and its
distance from the Sun is r. Express v1 in terms of r1 , r 2 and v2 .
65. A puck moves in a circle on the end of a string on a frictionless horizontal surface. The
magnitude of the puck’s angular momentum is L about the center of the circle. The string then
breaks and the puck begins€to travel in a straight
€ € line
€that is tangent to the circle at the point the
puck was when the string broke. What is true about the magnitude of the puck’s angular
momentum L ' about the center of the circle after the string breaks?
(a) L ' increases with time and is always larger than L
(b) L ' decreases with time and is always smaller than L
(c)€ L ' is constant and is equal to L
Explain.
€
€66. A man leaves a diving board rotating clockwise about his center of mass. Can the man do a
counterclockwise somersault before hitting the water? Explain.
€
67. A student is spinning on a stool holding a dumbbell in each outstretched arm. The student
suddenly drops the dumbbells. The angular speed of the student just after releasing the
dumbbells is (less than, equal to, greater than) his rotational speed just before releasing the
dumbbells. Explain.
68. A satellite is in circular orbit around Earth. The magnitude of the satellite’s angular momentum
about the center of the Earth can be found by multiplying the orbital radius by the magnitude of
the satellite’s ____.
69. Consider again the comet of problem 64. As the comet moves from position 2 to position 1
(a) the magnitude of its angular momentum about the Sun (increases, decreases, remains the
same).
(b) the linear kinetic energy of the comet (increases, decreases, remains the same).
70.
Initial State
5
4
3
2
1
0
1
2
3
4
5
A system consists of two point masses whose initial positions and velocities are shown. No
external forces act on the system. Collisions of a mass with the other mass or with the
boundary of the region in which the masses move are elastic. Five possible later states of this
system are shown below. Which of the later states are possible? If a later state is not possible
then explain why.
Later State A
Later State B
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
0
Later State C
1
2
3
4
5
Later State D
Later State E
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
71.
M
L
L
L
M
v
M
Two very small objects each have mass M and are connected by a rigid rod of length 3L and
negligible mass. A very small third object of mass M is traveling with speed v on a frictionless
horizontal surface as shown. The third object collides with the mass-rod-mass assembly and
sticks to it. Express your answers to the following in terms of M, L and v where appropriate.
(a) Where is the center of mass of the mass-rod-two mass assembly just after the collision?
Hint:
Suppose the mass-rod-two mass assembly were lying on the ground. If you lifted the
assembly by applying a force at the center of mass then the assembly would not rotate
because the torque due to the weight of one object would balance the torque due to the two
stuck together objects.
(b) Is linear momentum conserved in the collision? Why?
(c) Draw a vector at the center of mass of the mass-rod-two mass assembly showing its
direction of motion after the collision.
(d) What is the linear speed of the center of mass of the assembly just after the collision?
(e) Is angular momentum conserved in this collision? Why?
(f) What is the rotational inertia of the mass-rod-two mass assembly about its center of mass?
(g) What is the angular momentum of the system about the center of mass of the
mass-rod-two mass assembly just after the collision?
(h) What is the angular speed of the assembly about its center of mass just after the collision?
(i) Do you expect the total kinetic energy of the assembly after the collision to be less than,
equal to or greater than the kinetic energy of the third mass before the collision? Why?
Static Equilibrium Problems (Sections 33 to 35)
Use g= 9.80 m /s2 as needed.
72.
80.0 N
60.0 N
€
A
B
50.0 N
40.0 cm
weight= 120 N
A uniform bar AB is 2.00 m long and weighs 120 N. An upward force of 50.0 N is
€
applied 40.0 cm from end A. A downward
force of 60.0 N is applied at end A and a downward
force of 80.0 N is applied at end B. Find the magnitude, direction and location of the
equilibrant force. If the equilibrant force is applied to the bar then it will be in both translational
and rotational equilibrium.
€
73. A uniform board is carried by Willie the Wimp and Hank the Hunk. Willie holds up the left
end of the board. The board is held horizontally and is 6.0 m long. Where should Hank hold
the board so that he supports twice as much of its weight as Willie?
74. A non-uniform board is 7.00 m long. The center of gravity of the board is 3.00 m from the left
end. The mass of the board is 200 kg. The board is suspended horizontally by a rope at each
end. Find the tension in the left rope and the tension in the right rope.
€
75. A uniform bar 60 cm long is oriented vertically and acted upon by the forces shown. In which
of the four cases will the bar be in both translational and rotational equilibrium? Consider only
the horizontal forces shown.
€
60 cm
€
40 cm
€
60 N
30 cm
20€ cm
€
x newtons
y newtons
x
y
z
(a)
5.0
5.0
10
(b)
5.0
5.0
15
(c)
15
€ 10
35
(d)
35
15
10
€
€
€
€
10 cm
z newtons